Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot x = 0\}. $$ Equip $\mathbb R^6$ with the standard Euclidean metric and let $V$ inherit that metric from $\mathbb R^6$. Is there a sensible notion of the isomorphism class of the pair $(V, \Lambda)$? And how would this notion depend on the choice of the metric?
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1$\begingroup$ Let A be the order 4 identity matrix with a couple of zero columns adjoined. Then V looks like a 2-dimensional plane in the 6-space with the points having the first four coordinates all zero. The lattice looks like the appropriate sublattice. Use elementary row operations on A, and see how V and the sublattice change accordingly. If there is any nontrivial notion of isomorphism type, a combination of row operations will reveal it. Gerhard "Looks Like Planes To Me" Paseman, 2018.07.13. $\endgroup$Gerhard Paseman– Gerhard Paseman2018-07-13 16:19:51 +00:00Commented Jul 13, 2018 at 16:19
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